Unbounded symbols, heat flow, and Toeplitz operators
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914257813110784 |
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| author | Looi, Sam |
| author_facet | Looi, Sam |
| contents | We disprove the natural domain extension of the Berger--Coburn heat-flow conjecture for Toeplitz operators on the Bargmann space and identify the failure mechanism as a gap between pointwise and uniform control of a Gaussian averaging of the squared modulus of the symbol, a gap that is invisible to the linear form $T_g$. We establish that the form-defined operator $T_g$ and the natural-domain operator $U_g$ decouple in the unbounded symbols regime: while $T_g$ is governed by linear averaging, $U_g$ is controlled by the quadratic intensity of $|g|^2$. We construct a smooth, nonnegative radial symbol $g$ satisfying the coherent-state admissibility hypothesis with bounded heat transforms for all time $t>0$; for this symbol, $T_g$ is bounded, yet $U_g$ is unbounded. This is a strictly global phenomenon: under the coherent-state hypothesis, local singularities are insufficient to cause unboundedness, leaving the ``geometry at infinity'' as the sole obstruction. Boundedness of $U_g$ is equivalent to the condition that $|g|^2 dμ$ is a Fock--Carleson measure, a condition strictly stronger than the linear average $g dμ$ governing $T_g$. Finally, regarding the gap between the known sub-critical sufficiency condition and the critical heat time, we prove that heat-flow regularity is irreversible in this context and show that bootstrapping strategies cannot resolve the gap between sufficiency and critical time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_10711 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Unbounded symbols, heat flow, and Toeplitz operators Looi, Sam Functional Analysis Analysis of PDEs Complex Variables We disprove the natural domain extension of the Berger--Coburn heat-flow conjecture for Toeplitz operators on the Bargmann space and identify the failure mechanism as a gap between pointwise and uniform control of a Gaussian averaging of the squared modulus of the symbol, a gap that is invisible to the linear form $T_g$. We establish that the form-defined operator $T_g$ and the natural-domain operator $U_g$ decouple in the unbounded symbols regime: while $T_g$ is governed by linear averaging, $U_g$ is controlled by the quadratic intensity of $|g|^2$. We construct a smooth, nonnegative radial symbol $g$ satisfying the coherent-state admissibility hypothesis with bounded heat transforms for all time $t>0$; for this symbol, $T_g$ is bounded, yet $U_g$ is unbounded. This is a strictly global phenomenon: under the coherent-state hypothesis, local singularities are insufficient to cause unboundedness, leaving the ``geometry at infinity'' as the sole obstruction. Boundedness of $U_g$ is equivalent to the condition that $|g|^2 dμ$ is a Fock--Carleson measure, a condition strictly stronger than the linear average $g dμ$ governing $T_g$. Finally, regarding the gap between the known sub-critical sufficiency condition and the critical heat time, we prove that heat-flow regularity is irreversible in this context and show that bootstrapping strategies cannot resolve the gap between sufficiency and critical time. |
| title | Unbounded symbols, heat flow, and Toeplitz operators |
| topic | Functional Analysis Analysis of PDEs Complex Variables |
| url | https://arxiv.org/abs/2601.10711 |